Moment-generating functions

In statistics, a moment-generating function (MGF) is an alternative to the Statistical distributions for describing a probability distribution. For random variable , its MGF is given as

because -th derivative of this function, evaluated at , returns the -th moment of X about .

Evaluating for , we obtain the -th moment about 0:

Sources

1. Wikipedia: Moment-generating function
2. Gundersen 2020